Ergodicity conditions for two-dimensional Markov chains on the positive quadrant

Summary The basic problem considered in this paper is that of determining conditions for recurrence and transience for two dimensional irreducible Markov chains whose state space is Z ₊ ² =Z₊xZ₊. Assuming bounded jumps and a homogeneity condition Malyshev [7] obtained necessary and sufficient conditions for recurrence and transience of two dimensional random walks on the positive quadrant. Unfortunately, his hypothesis that the jumps of the Markov chain be bounded rules out for example, the Poisson arrival process. In this paper we generalise Malyshev's theorem by means of a method that makes novel use of the solution to Laplace's equation in the first quadrant satisfying an oblique derivative condition on the boundaries. This method, which allows one to replace the very restrictive boundedness condition by a moment condition and a lower boundedness condition, is of independent interest.     

A continuity theorem in the ruin problem

For a random walk, we prove a continuity theorem for the exit time from a strip containing the abscissa axis, including the case of the exit time from a strip through an a priori chosen boundary. In particular, we compute the ruin probabilities for two classes of centered distributions.     

Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem

We consider in this paper that the reserve of an insurance company follows the classical model, in which the aggregate claim amount follows a compound Poisson process. Our goal is to minimize the ruin probability of the company assuming that the management can invest dynamically part of the reserve in an asset that has a positive fixed return. However, due to transaction costs, the sale price of the asset at the time when the company needs cash to cover claims is lower than the original price. This is a singular two-dimensional stochastic control problem which cannot be reduced to a one-dimensional problem. The associated Hamilton–Jacobi–Bellman (HJB) equation is a variational inequality involving a first order integro-differential operator and a gradient constraint. We characterize the optimal value function as the unique viscosity solution of the associated HJB equation. For exponential claim distributions, we show that the optimal value function is induced by a two-region stationary strategy (“action” and “inaction” regions) and we find an implicit formula for the free boundary between these two regions. We also study the optimal strategy for small and large initial capital and show some numerical examples.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.     

Validation of positive quadrant dependence

Quadrant dependence is a useful dependence notion of two random variables, widely applied in reliability, insurance and actuarial sciences. The interest in this dependence structure ranges from modeling it, throughout measuring its strength and investigations on how increasing the dependence effects of several reliability and economic indexes, to hypothesis testing on the dependence. In this paper, we focus on testing for positive quadrant dependence. We propose two new tests for verifying positive quadrant dependence. We prove novel results on finite sample behavior of power function of one of the proposed tests as well as evaluate and compare the two new solutions with the best existing ones, via a simulation study. These comparisons demonstrate that the new solutions are slightly weaker in detecting positive quadrant dependence modeled by classical bivariate models and outperform the best existing solutions when some mixtures, regression and heavy-tailed models have to be detected. Finally, the methods introduced in the paper are applied to real life insurance data, to assess the dependence and test them for positive quadrant dependence.     

A note on the two-sided two-dimensional moment problem

It is shown that there is a positive semidefinite two-sided two-dimensional sequence f, which is not a moment sequence, such that log f(2m,2n) =O(m ² + n ²) as (m,n) →∞. This revised version was published online in June 2006 with corrections to the Cover Date.     

二维风险模型中Copula 相依下的破产概率

在本文中, 我们把Copula 连结函数用到二维的风险模型中, 考虑两个模型索赔额之间基于Copula 的相依关系. 首先对二维复合Poisson 模型给出了最早破产时刻定义下的生存概率满足的偏微分方程; 然后对二维的复合二项模型, 分别在连续型索赔额分布和离散型索赔额分布下给出了不同定义的生存概率和破产概率的递归公式, 并且特别选择了FGM Copula 连结函数, 给出了相应的结果; 另外在离散型分布下, 对于其Copula 函数的不唯一性进行了说明.     

一个特殊的二维除数问题

设△*(a,b;x)为D*(a,b;x)=∑manb≤x(m,n)=11,(1≤a＜b,(a,b)=1)的余项.本文在黎曼假设下利用指数和方法获得了△*(a,b;x)上界估计的一个较好估计.     

The classical ruin problem on a finite solvable group

For the classical ruin problem (a special case of a cyclic group), we use an explicit expression for the characteristic function of the time of first hitting an arbitrary subset of a finite solvable group by a random walk, from a fixed subset, to obtain a new proof of the well-known formula which allows one to estimate the characteristic function of the ruin probability (hitting the identity of a group) at theth- trial.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 361–366, March, 1993.     

A further note on a two-dimensional free-boundary problem

Zusammenfassung Diese Notiz enthält eine numerische Illustration zur VeröffentlichungOn a Two-Dimensional Free-Boundary Problem. Dabei wird vermittelst numerischer Integration der in der erwähnten Schrift vorkommenden Differentialgleichung der Fall eines Kanals mit einer Krümmung von 90° behandelt. Es ist beabsichtigt, zu einem späteren Zeitpunkt eine umfassendere Darstellung der Lösungen dieser Differentialgleichung zu veröffentlichen.     

经典风险模型下带有随机利率的一类破产问题

考虑利率随机性通过标准布朗运动和普哇松过程来描述情形下的一类破产问题．利用鞅方法,得到了此情形下经典风险模型的Lundberg基本方程,并考虑了其解的两个有效应用,从而得到了破产概率、盈余首次到达某给定水平x（x〉u）的概率、f（x,y｜0）及初始盈余u=0情况下破产时单位赔付现值的表达式．最后给出了当个体理赔服从指数分布情形下的一些结果．     

A positive answer to the basis problem

lcub;x _n rcub; with lcub;x _n ,x* _n rcub; biorthogonal is a “uniformly minimal basis with quasifixed brackets and permutations” of a Banach spaceX if lcub;x _n rcub; andx* _n rcub; are both bounded. Moreover, there is an increasing sequence lcub;q _m rcub; of positive integers such that, for eachx′ ofX, settingq′(0)=0, $$x' = \sum\limits_{m = 0}^\infty { \sum\limits_{n = q'(m) + 1}^{q'(m + 1)} {x_{\pi '(n)}^ * (x')x_{\pi '(n)} ,} } $$ , where, for eachm≥1,q(m)+1≤q′(m)≤q(m+1) while $$\left\{ {\pi '(n)} \right\}_{n = q(m) + 1}^{q(m + 1)} is a permutation of \left\{ n \right\}_{n = q(m) + 1}^{q(m + 1)} .$$ . Then, for each subspaceY of a separable Banach spaceX, there exists a uniformly minimal basis with quasi-fixed brackets and permutations ofY, which can be extended to a uniformly minimal basis with quasi-fixed brackets and permutations ofX.     

A two-dimensional inverse problem for the viscoelasticity equation

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ?² | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.     

A two-dimensional glimm type scheme on cauchy problem of two-dimensional scalar conservation law

In this paper,we construct a new two-dimensional convergent scheme to solve Cauchy problem of following two-dimensional scalar conservation law{?_tu +?_xf(u) + ?_yg(u) = 0,u(x,y,0) = u_0(x,y).In which initial data can be unbounded.Although the existence and uniqueness of the weak entropy solution are obtained,little is known about how to investigate two-dimensional or higher dimensional conservation law by the schemes based on wave interaction of 2D Riemann solutions and their estimation.So we construct such scheme in our paper and get some new results.     

The gambler’s ruin problem with delays

A recent paper by  Pozdnyakov and Steele (2010) is devoted to the so-called binary-plus-passive design. Two problems that the authors do not consider can be identified with the classical gambler’s ruin problem in which delays are allowed.